The Mathematical Bases for Qualitative Reasoning

Abstract: Much reasoning about quantities takes place without use of mathematical
formalisms, but solely in terms of ordinary language. A good deal of such
qualitative reasoning makes implicit use of the properties of ordinal
variables and monotonic transformations. In this paper, we attempt to provide
the formal foundations of qualitative analysis and to show how qualitative
reasoning arises naturally and simply out of the structure of systems of
algebraic equations and ordinary differential equations. Our goal is to
explicate, using familiar mathematical formalisms, the practices of
researchers in many fields who use qualitative reasoning, and therby to gain
an understanding of the formal assumptions and mechanisms that underlie such
analysis. We sketch out some of the properties of functions, and especially
continuous differentiable functions, that are invariant under monotonic
transformations of the variables, and show how these properties can be used to
analyze phenomena where the variables are only defined ordinally.